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K-mean

head(iris)

plot(iris[,3:4])

irisCluster <- kmeans(iris[, 3:4], 3, nstart = 20)
irisCluster$cluster <- as.factor(irisCluster$cluster)
library(ggplot2)

ggplot(iris, aes(Petal.Length, Petal.Width, col = irisCluster$cluster, fill=Species)) + 
  stat_ellipse(geom = "polygon", col = "black", alpha = 0.5) +
  geom_point()

k.max <- 15
wss <- sapply(1:k.max, function(k){kmeans(iris[, 3:4], k, nstart=50, iter.max = 15)$tot.withinss})

plot(1:k.max, wss,
     type="b", pch = 19, frame = FALSE, 
     xlab="Number of clusters K",
     ylab="Total within-clusters sum of squares")

Periodogram

counter <- 500
x <- seq(counter)

y <- 2 * cos(2 * pi * (1/50) * x + 0.6 * pi)
signal <- y + rnorm(counter)
plot(x, y, type = "l")

plot(x, signal, type = "l")

sunspots <- scan("/Users/ansonleung/Desktop/Applied Financial Engineering/Financial Times Series Analysis/Lecture 3/sunspots.dat")

Read 459 items

plot(sunspots, type="b")

x = diff(sunspots)

I = abs(fft(x)/sqrt(458))^2
P = (4/458)*I[1:230]

freq = (0:229)/458

plot(freq, P, type="l")

library(astsa)

x <- scan("/Users/ansonleung/Desktop/Applied Financial Engineering/Financial Times Series Analysis/Lecture 3/recruit.dat")

Read 453 items

mvspec(x, log="no")

k = kernel("daniell", 4)
mvspec(x, k, log="no")

k = kernel("daniell", c(4,4)) 
mvspec(x, k, log="no")

specvalues = mvspec(x, k, log="no")

specvalues$details

       frequency   period  spectrum
  [1,]    0.0021 480.0000 3383.5058
  [2,]    0.0042 240.0000 3723.6683
  [3,]    0.0063 160.0000 4063.3578
  [4,]    0.0083 120.0000 4498.0099
  [5,]    0.0104  96.0000 4948.5148
  [6,]    0.0125  80.0000 5497.4841
  [7,]    0.0146  68.5714 5948.6416
  [8,]    0.0167  60.0000 6450.8076
  [9,]    0.0187  53.3333 6840.4396
 [10,]    0.0208  48.0000 7154.9638
 [11,]    0.0229  43.6364 7198.2791
 [12,]    0.0250  40.0000 7093.4142
 [13,]    0.0271  36.9231 6889.4382
 [14,]    0.0292  34.2857 6761.5608
 [15,]    0.0312  32.0000 6389.7518
 [16,]    0.0333  30.0000 6060.4244
 [17,]    0.0354  28.2353 5720.6943
 [18,]    0.0375  26.6667 5252.6571
 [19,]    0.0396  25.2632 4727.8013
 [20,]    0.0417  24.0000 4252.4430
 [21,]    0.0437  22.8571 3848.0378
 [22,]    0.0458  21.8182 3493.5831
 [23,]    0.0479  20.8696 2963.9326
 [24,]    0.0500  20.0000 2572.4034
 [25,]    0.0521  19.2000 2161.3083
 [26,]    0.0542  18.4615 1700.2617
 [27,]    0.0562  17.7778 1389.7388
 [28,]    0.0583  17.1429 1146.9481
 [29,]    0.0604  16.5517  964.3456
 [30,]    0.0625  16.0000  823.3735
 [31,]    0.0646  15.4839  686.7090
 [32,]    0.0667  15.0000  897.8008
 [33,]    0.0688  14.5455 1103.4691
 [34,]    0.0708  14.1176 1328.1553
 [35,]    0.0729  13.7143 1607.2885
 [36,]    0.0750  13.3333 1837.4561
 [37,]    0.0771  12.9730 2074.8114
 [38,]    0.0792  12.6316 2302.5845
 [39,]    0.0813  12.3077 2516.8509
 [40,]    0.0833  12.0000 2746.4671
 [41,]    0.0854  11.7073 2461.0304
 [42,]    0.0875  11.4286 2177.9664
 [43,]    0.0896  11.1628 1898.9515
 [44,]    0.0917  10.9091 1623.8459
 [45,]    0.0938  10.6667 1384.6294
 [46,]    0.0958  10.4348 1137.8088
 [47,]    0.0979  10.2128  899.7337
 [48,]    0.1000  10.0000  661.3318
 [49,]    0.1021   9.7959  398.7428
 [50,]    0.1042   9.6000  401.7536
 [51,]    0.1063   9.4118  402.7817
 [52,]    0.1083   9.2308  401.9023
 [53,]    0.1104   9.0566  392.1142
 [54,]    0.1125   8.8889  359.3417
 [55,]    0.1146   8.7273  324.0378
 [56,]    0.1167   8.5714  289.3280
 [57,]    0.1188   8.4211  258.3219
 [58,]    0.1208   8.2759  242.3243
 [59,]    0.1229   8.1356  226.1004
 [60,]    0.1250   8.0000  214.9562
 [61,]    0.1271   7.8689  210.3355
 [62,]    0.1292   7.7419  211.2108
 [63,]    0.1312   7.6190  224.6399
 [64,]    0.1333   7.5000  237.9108
 [65,]    0.1354   7.3846  250.1213
 [66,]    0.1375   7.2727  265.0379
 [67,]    0.1396   7.1642  270.3521
 [68,]    0.1417   7.0588  261.7040
 [69,]    0.1437   6.9565  244.2570
 [70,]    0.1458   6.8571  218.1679
 [71,]    0.1479   6.7606  190.8703
 [72,]    0.1500   6.6667  183.1164
 [73,]    0.1521   6.5753  180.4624
 [74,]    0.1542   6.4865  182.3508
 [75,]    0.1562   6.4000  186.0763
 [76,]    0.1583   6.3158  195.4318
 [77,]    0.1604   6.2338  214.3466
 [78,]    0.1625   6.1538  240.0628
 [79,]    0.1646   6.0759  267.9256
 [80,]    0.1667   6.0000  296.1027
 [81,]    0.1687   5.9259  284.9247
 [82,]    0.1708   5.8537  270.5449
 [83,]    0.1729   5.7831  248.7994
 [84,]    0.1750   5.7143  221.6484
 [85,]    0.1771   5.6471  195.8172
 [86,]    0.1792   5.5814  170.1639
 [87,]    0.1812   5.5172  141.6886
 [88,]    0.1833   5.4545  114.6808
 [89,]    0.1854   5.3933   86.7627
 [90,]    0.1875   5.3333   79.8203
 [91,]    0.1896   5.2747   73.7594
 [92,]    0.1917   5.2174   70.9909
 [93,]    0.1937   5.1613   68.8111
 [94,]    0.1958   5.1064   63.6917
 [95,]    0.1979   5.0526   58.3660
 [96,]    0.2000   5.0000   54.4895
 [97,]    0.2021   4.9485   50.4913
 [98,]    0.2042   4.8980   47.7787
 [99,]    0.2062   4.8485   43.3229
[100,]    0.2083   4.8000   39.8791
[101,]    0.2104   4.7525   36.2312
[102,]    0.2125   4.7059   34.4717
[103,]    0.2146   4.6602   34.1750
[104,]    0.2167   4.6154   35.9153
[105,]    0.2188   4.5714   37.6983
[106,]    0.2208   4.5283   39.0949
[107,]    0.2229   4.4860   40.4487
[108,]    0.2250   4.4444   41.2742
[109,]    0.2271   4.4037   41.9460
[110,]    0.2292   4.3636   43.1031
[111,]    0.2312   4.3243   44.1189
[112,]    0.2333   4.2857   48.0360
[113,]    0.2354   4.2478   48.5140
[114,]    0.2375   4.2105   49.8358
[115,]    0.2396   4.1739   51.4480
[116,]    0.2417   4.1379   52.1465
[117,]    0.2438   4.1026   53.8571
[118,]    0.2458   4.0678   54.2340
[119,]    0.2479   4.0336   54.5173
[120,]    0.2500   4.0000   54.5018
[121,]    0.2521   3.9669   49.3128
[122,]    0.2542   3.9344   46.0916
[123,]    0.2562   3.9024   41.1002
[124,]    0.2583   3.8710   36.2660
[125,]    0.2604   3.8400   31.9392
[126,]    0.2625   3.8095   27.6790
[127,]    0.2646   3.7795   24.6577
[128,]    0.2667   3.7500   21.4706
[129,]    0.2688   3.7209   18.9509
[130,]    0.2708   3.6923   18.8600
[131,]    0.2729   3.6641   18.8698
[132,]    0.2750   3.6364   19.9140
[133,]    0.2771   3.6090   21.0819
[134,]    0.2792   3.5821   23.3498
[135,]    0.2812   3.5556   25.4372
[136,]    0.2833   3.5294   27.7179
[137,]    0.2854   3.5036   30.2683
[138,]    0.2875   3.4783   32.2368
[139,]    0.2896   3.4532   33.8993
[140,]    0.2917   3.4286   34.4704
[141,]    0.2938   3.4043   34.6390
[142,]    0.2958   3.3803   34.5751
[143,]    0.2979   3.3566   32.5440
[144,]    0.3000   3.3333   30.7099
[145,]    0.3021   3.3103   28.2449
[146,]    0.3042   3.2877   26.0939
[147,]    0.3062   3.2653   24.6177
[148,]    0.3083   3.2432   23.7187
[149,]    0.3104   3.2215   23.9083
[150,]    0.3125   3.2000   24.7318
[151,]    0.3146   3.1788   25.6631
[152,]    0.3167   3.1579   27.6031
[153,]    0.3188   3.1373   29.7384
[154,]    0.3208   3.1169   31.8160
[155,]    0.3229   3.0968   32.9871
[156,]    0.3250   3.0769   34.3741
[157,]    0.3271   3.0573   35.3200
[158,]    0.3292   3.0380   35.5154
[159,]    0.3312   3.0189   34.8757
[160,]    0.3333   3.0000   34.1192
[161,]    0.3354   2.9814   33.1808
[162,]    0.3375   2.9630   31.5040
[163,]    0.3396   2.9448   29.6633
[164,]    0.3417   2.9268   28.9341
[165,]    0.3438   2.9091   27.0342
[166,]    0.3458   2.8916   25.3162
[167,]    0.3479   2.8743   23.8679
[168,]    0.3500   2.8571   22.9800
[169,]    0.3521   2.8402   22.1128
[170,]    0.3542   2.8235   20.9879
[171,]    0.3562   2.8070   20.8163
[172,]    0.3583   2.7907   21.1241
[173,]    0.3604   2.7746   20.4429
[174,]    0.3625   2.7586   20.1445
[175,]    0.3646   2.7429   20.2149
[176,]    0.3667   2.7273   20.4241
[177,]    0.3687   2.7119   20.7374
[178,]    0.3708   2.6966   21.0242
[179,]    0.3729   2.6816   21.4956
[180,]    0.3750   2.6667   21.2115
[181,]    0.3771   2.6519   20.8832
[182,]    0.3792   2.6374   20.5005
[183,]    0.3812   2.6230   20.2490
[184,]    0.3833   2.6087   19.6282
[185,]    0.3854   2.5946   19.0138
[186,]    0.3875   2.5806   18.2154
[187,]    0.3896   2.5668   17.5413
[188,]    0.3917   2.5532   17.2190
[189,]    0.3938   2.5397   17.0795
[190,]    0.3958   2.5263   16.8594
[191,]    0.3979   2.5131   17.0401
[192,]    0.4000   2.5000   17.3831
[193,]    0.4021   2.4870   17.5899
[194,]    0.4042   2.4742   17.7111
[195,]    0.4062   2.4615   17.4108
[196,]    0.4083   2.4490   17.2912
[197,]    0.4104   2.4365   16.7260
[198,]    0.4125   2.4242   16.3159
[199,]    0.4146   2.4121   15.8501
[200,]    0.4167   2.4000   15.4962
[201,]    0.4188   2.3881   14.6751
[202,]    0.4208   2.3762   13.9146
[203,]    0.4229   2.3645   13.1885
[204,]    0.4250   2.3529   13.2600
[205,]    0.4271   2.3415   12.7958
[206,]    0.4292   2.3301   12.5930
[207,]    0.4312   2.3188   12.8403
[208,]    0.4333   2.3077   13.1415
[209,]    0.4354   2.2967   13.2417
[210,]    0.4375   2.2857   14.0035
[211,]    0.4396   2.2749   14.8994
[212,]    0.4417   2.2642   15.9015
[213,]    0.4438   2.2535   16.0433
[214,]    0.4458   2.2430   16.5519
[215,]    0.4479   2.2326   17.0329
[216,]    0.4500   2.2222   16.0607
[217,]    0.4521   2.2120   15.3824
[218,]    0.4542   2.2018   14.8945
[219,]    0.4562   2.1918   13.8862
[220,]    0.4583   2.1818   13.0781
[221,]    0.4604   2.1719   12.3491
[222,]    0.4625   2.1622   12.0749
[223,]    0.4646   2.1525   11.9012
[224,]    0.4667   2.1429   11.7565
[225,]    0.4688   2.1333   12.5923
[226,]    0.4708   2.1239   13.1748
[227,]    0.4729   2.1145   13.5462
[228,]    0.4750   2.1053   14.0356
[229,]    0.4771   2.0961   14.2253
[230,]    0.4792   2.0870   14.0273
[231,]    0.4812   2.0779   13.6965
[232,]    0.4833   2.0690   13.1697
[233,]    0.4854   2.0601   12.5730
[234,]    0.4875   2.0513   11.8617
[235,]    0.4896   2.0426   11.0409
[236,]    0.4917   2.0339   10.3763
[237,]    0.4937   2.0253    9.7607
[238,]    0.4958   2.0168    9.3193
[239,]    0.4979   2.0084    9.1320
[240,]    0.5000   2.0000    9.0994

Cluster analysis

mydata <- read.csv("/Users/ansonleung/Desktop/Applied Financial Engineering/Financial Times Series Analysis/Lecture 3/utilities.csv")
head(mydata)

str(mydata)

'data.frame':   22 obs. of  9 variables:
 $ Company     : chr  "Arizona " "Boston " "Central " "Commonwealth" ...
 $ Fixed_charge: num  1.06 0.89 1.43 1.02 1.49 1.32 1.22 1.1 1.34 1.12 ...
 $ RoR         : num  9.2 10.3 15.4 11.2 8.8 13.5 12.2 9.2 13 12.4 ...
 $ Cost        : int  151 202 113 168 192 111 175 245 168 197 ...
 $ Load        : num  54.4 57.9 53 56 51.2 60 67.6 57 60.4 53 ...
 $ D.Demand    : num  1.6 2.2 3.4 0.3 1 -2.2 2.2 3.3 7.2 2.7 ...
 $ Sales       : int  9077 5088 9212 6423 3300 11127 7642 13082 8406 6455 ...
 $ Nuclear     : num  0 25.3 0 34.3 15.6 22.5 0 0 0 39.2 ...
 $ Fuel_Cost   : num  0.628 1.555 1.058 0.7 2.044 ...

pairs(mydata[,-c(1,1)])

plot(mydata$Fuel_Cost~ mydata$Sales, data = mydata)

with(mydata,
     text(mydata$Fuel_Cost ~ mydata$Sales, labels=mydata$Company,pos=4, cex = 0.4)
     )

z = mydata[,-c(1,1)]

means = apply(z,2,mean)
sds = apply(z,2,sd)
nor = scale(z,center=means,scale=sds)

distance = dist(nor)

print(distance, digits = 3)

      1    2    3    4    5    6    7    8    9   10   11   12   13   14   15   16   17   18   19   20   21
2  3.10                                                                                                    
3  3.68 4.92                                                                                               
4  2.46 2.16 4.11                                                                                          
5  4.12 3.85 4.47 4.13                                                                                     
6  3.61 4.22 2.99 3.20 4.60                                                                                
7  3.90 3.45 4.22 3.97 4.60 3.35                                                                           
8  2.74 3.89 4.99 3.69 5.16 4.91 4.36                                                                      
9  3.25 3.96 2.75 3.75 4.49 3.73 2.80 3.59                                                                 
10 3.10 2.71 3.93 1.49 4.05 3.83 4.51 3.67 3.57                                                            
11 3.49 4.79 5.90 4.86 6.46 6.00 6.00 3.46 5.18 5.08                                                       
12 3.22 2.43 4.03 3.50 3.60 3.74 1.66 4.06 2.74 3.94 5.21                                                  
13 3.96 3.43 4.39 2.58 4.76 4.55 5.01 4.14 3.66 1.41 5.31 4.50                                             
14 2.11 4.32 2.74 3.23 4.82 3.47 4.91 4.34 3.82 3.61 4.32 4.34 4.39                                        
15 2.59 2.50 5.16 3.19 4.26 4.07 2.93 3.85 4.11 4.26 4.74 2.33 5.10 4.24                                   
16 4.03 4.84 5.26 4.97 5.82 5.84 5.04 2.20 3.63 4.53 3.43 4.62 4.41 5.17 5.18                              
17 4.40 3.62 6.36 4.89 5.63 6.10 4.58 5.43 4.90 5.48 4.75 3.50 5.61 5.56 3.40 5.56                         
18 1.88 2.90 2.72 2.65 4.34 2.85 2.95 3.24 2.43 3.07 3.95 2.45 3.78 2.30 3.00 3.97 4.43                    
19 2.41 4.63 3.18 3.46 5.13 2.58 4.52 4.11 4.11 4.13 4.52 4.41 5.01 1.88 4.03 5.23 6.09 2.47               
20 3.17 3.00 3.73 1.82 4.39 2.91 3.54 4.09 2.95 2.05 5.35 3.43 2.23 3.74 3.78 4.82 4.87 2.92 3.90          
21 3.45 2.32 5.09 3.88 3.64 4.63 2.68 3.98 3.74 4.36 4.88 1.38 4.94 4.93 2.10 4.57 3.10 3.19 4.97 4.15     
22 2.51 2.42 4.11 2.58 3.77 4.03 4.00 3.24 3.21 2.56 3.44 3.00 2.74 3.51 3.35 3.46 3.63 2.55 3.97 2.62 3.01

mydata.hclust = hclust(distance)
plot(mydata.hclust)

plot(mydata.hclust,labels=mydata$Company,main='Default from hclust')

plot(mydata.hclust,hang=-1)

member = cutree(mydata.hclust,3)

table(member)

member
 1  2  3 
14  5  3 

aggregate(nor,list(member),mean)

aggregate(mydata[,-c(1,1)],list(member),mean)

wss <- (nrow(nor)-1)*sum(apply(nor,2,var))
for (i in 2:20) wss[i] <- sum(kmeans(nor, centers=i)$withinss)
plot(1:20, wss, type="b", xlab="Number of Clusters", ylab="Within groups sum of squares")

kc <- kmeans(z,3)
plot(Sales~D.Demand, mydata, col = kc$cluster)

fractional R: Fractional Differences:

varve = scan("/Users/ansonleung/Desktop/Applied Financial Engineering/Financial Times Series Analysis/Lecture 3/varve.dat")

Read 634 items

varve = ts(varve)

library(arfima)

y = log(varve) - mean(log(varve))

acf(y)

varvefd = arfima(y)

Note: only one starting point.  Only one mode can be found -- this is now the default behavior.
Beginning the fits with 1 starting values.

d = summary(varvefd)$coef[[1]][1]

resids = resid(varvefd)[[1]] 

plot.ts(resids)

acf(resids)

Looping through p and q

data(Mishkin,package="Ecdat")

x <- diff(as.vector(Mishkin[,1]))

plot(x, type = 'l')

result <- matrix(0, nrow=9, ncol=4)
idx <- 1

for (i in 0:2)
{
  for (j in 0:2){
    fit = arima(x,order=c(i,0,j))
    result[idx, 1] = i
    result[idx, 2] = j
    result[idx, 3] = fit$aic
    result[idx, 4] = result[idx,3] + (log(length(x))-2)*i
    idx = idx + 1
    }
}
result <- data.frame(result)
names(result) <- c('p', 'q', 'AIC', 'BIC')
result

tar model

flu = scan("/Users/ansonleung/Desktop/Applied Financial Engineering/Financial Times Series Analysis/Lecture 3/flu.dat")

Read 132 items

flu = ts(flu)

plot(flu,type="b")

y = diff(flu,1)
plot(y,type="b")

model = ts.intersect(y, lag1y=lag(y,-1), lag2y=lag(y, -2), lag3y=lag(y,-3), lag4y=lag(y, -4))

x = model[,1]
P = model[,2:5]
c = 0.05 ## Threshold value

less = (P[,1]<c)
x1 = x[less]
P1 = P[less,]
out1 = lm(x1~P1[,1]+P1[,2]+P1[,3]+P1[,4])
summary(out1)

Call:
lm(formula = x1 ~ P1[, 1] + P1[, 2] + P1[, 3] + P1[, 4])

Residuals:
     Min       1Q   Median       3Q      Max 
-0.13312 -0.02049  0.00218  0.01667  0.26666 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.004471   0.004894   0.914 0.363032    
P1[, 1]      0.506650   0.078319   6.469  3.2e-09 ***
P1[, 2]     -0.200086   0.056573  -3.537 0.000604 ***
P1[, 3]      0.121047   0.054463   2.223 0.028389 *  
P1[, 4]     -0.110938   0.045979  -2.413 0.017564 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.04578 on 105 degrees of freedom
Multiple R-squared:  0.3763,    Adjusted R-squared:  0.3526 
F-statistic: 15.84 on 4 and 105 DF,  p-value: 3.568e-10

greater = (P[,1]>=c)
x2 = x[greater]
P2 = P[greater,]
out2 = lm(x2~P2[,1]+P2[,2]+P2[,3]+P2[,4])
summary(out2)

Call:
lm(formula = x2 ~ P2[, 1] + P2[, 2] + P2[, 3] + P2[, 4])

Residuals:
      Min        1Q    Median        3Q       Max 
-0.089975 -0.036825 -0.006328  0.040765  0.129509 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.40794    0.04675   8.726 1.53e-06 ***
P2[, 1]     -0.74833    0.16644  -4.496 0.000732 ***
P2[, 2]     -1.03231    0.21137  -4.884 0.000376 ***
P2[, 3]     -2.04504    1.05000  -1.948 0.075235 .  
P2[, 4]     -6.71178    1.24538  -5.389 0.000163 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.0721 on 12 degrees of freedom
Multiple R-squared:  0.9207,    Adjusted R-squared:  0.8943 
F-statistic: 34.85 on 4 and 12 DF,  p-value: 1.618e-06

res1 = residuals(out1)
res2 = residuals(out2)
less[less==1] = res1
greater[greater==1] = res2
resid = less + greater
acf(resid)

less = (P[,1]<c)
greater = (P[,1]>=c)
fit1 = predict(out1)
fit2 = predict(out2)
less[less==1]= fit1
greater[greater==1] = fit2
fit = less + greater
plot(y, type="o")
lines(fit, col = "red", lty="dashed")

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library(tsDyn)
dflu = diff(flu,1)
flu.tar4.05 = setar(dflu, m=4, thDelay=0, th=.05)

With the threshold you gave (0.05) there is a regime with less than trim=15% observations (86.61%, 13.39%, )

 1 T: Trim not respected:  0.8661417 0.1338583 from th: 0.05

Possible unit root in the high  regime. Roots are: 0.6182 0.6244 0.6244 0.6182

summary(flu.tar4.05)

Non linear autoregressive model

SETAR model ( 2 regimes)
Coefficients:
Low regime:
     const.L       phiL.1       phiL.2       phiL.3       phiL.4 
 0.004471044  0.506649694 -0.200086031  0.121047354 -0.110938271 

High regime:
   const.H     phiH.1     phiH.2     phiH.3     phiH.4 
 0.4079353 -0.7483325 -1.0323129 -2.0450407 -6.7117769 

Threshold:
-Variable: Z(t) = + (1) X(t)+ (0)X(t-1)+ (0)X(t-2)+ (0)X(t-3)
-Value: 0.05 (fixed)
Proportion of points in low regime: 86.61%   High regime: 13.39% 

Residuals:
       Min         1Q     Median         3Q        Max 
-0.1331155 -0.0215013  0.0021557  0.0197805  0.2666619 

Fit:
residuals variance = 0.002156,  AIC = -784, MAPE = 404.3%

Coefficient(s):

          Estimate  Std. Error  t value  Pr(>|t|)    
const.L  0.0044710   0.0051648   0.8657  0.388381    
phiL.1   0.5066497   0.0826526   6.1299 1.137e-08 ***
phiL.2  -0.2000860   0.0597035  -3.3513  0.001073 ** 
phiL.3   0.1210474   0.0574764   2.1060  0.037268 *  
phiL.4  -0.1109383   0.0485235  -2.2863  0.023977 *  
const.H  0.4079353   0.0313281  13.0214 < 2.2e-16 ***
phiH.1  -0.7483325   0.1115341  -6.7094 6.658e-10 ***
phiH.2  -1.0323129   0.1416409  -7.2882 3.525e-11 ***
phiH.3  -2.0450407   0.7036312  -2.9064  0.004349 ** 
phiH.4  -6.7117769   0.8345586  -8.0423 6.771e-13 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Threshold
Variable: Z(t) = + (1) X(t) + (0) X(t-1)+ (0) X(t-2)+ (0) X(t-3)

Value: 0.05 (fixed)

plot(flu.tar4.05)

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flu.tar4 = setar(dflu, m=4, thDelay=0)

Possible unit root in the high  regime. Roots are: 0.8074 0.8074 1.3596 1.3596

summary(flu.tar4)

Non linear autoregressive model

SETAR model ( 2 regimes)
Coefficients:
Low regime:
      const.L        phiL.1        phiL.2        phiL.3        phiL.4 
 0.0006269563  0.4608089284 -0.2243720404  0.1100931813 -0.1307031988 

High regime:
   const.H     phiH.1     phiH.2     phiH.3     phiH.4 
 0.2035231 -0.4071318 -1.4686776  0.3768388 -0.8298225 

Threshold:
-Variable: Z(t) = + (1) X(t)+ (0)X(t-1)+ (0)X(t-2)+ (0)X(t-3)
-Value: 0.03646
Proportion of points in low regime: 84.25%   High regime: 15.75% 

Residuals:
       Min         1Q     Median         3Q        Max 
-0.2632248 -0.0195074  0.0044446  0.0210898  0.2854727 

Fit:
residuals variance = 0.003739,  AIC = -710, MAPE = 541.4%

Coefficient(s):

           Estimate  Std. Error  t value  Pr(>|t|)    
const.L  0.00062696  0.00699184   0.0897  0.928698    
phiL.1   0.46080893  0.11083474   4.1576 6.035e-05 ***
phiL.2  -0.22437204  0.07882706  -2.8464  0.005196 ** 
phiL.3   0.11009318  0.07608021   1.4471  0.150464    
phiL.4  -0.13070320  0.06406509  -2.0402  0.043511 *  
const.H  0.20352307  0.02632410   7.7314 3.505e-12 ***
phiH.1  -0.40713175  0.13413815  -3.0352  0.002944 ** 
phiH.2  -1.46867761  0.17439989  -8.4213 8.911e-14 ***
phiH.3   0.37683878  0.75269827   0.5007  0.617527    
phiH.4  -0.82982249  0.84051869  -0.9873  0.325478    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Threshold
Variable: Z(t) = + (1) X(t) + (0) X(t-1)+ (0) X(t-2)+ (0) X(t-3)

Value: 0.03646

plot(flu.tar4)

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